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Derivatives and Differential Equations

Derivatives and Differential Equations. Differentiation. Differential change . Derivative Definition. Taylor Series. Taylor Series Graphically. Numerical Differentiation Based on Taylor Series . Forward finite-divided difference B ackward finite-divided difference

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Derivatives and Differential Equations

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  1. Derivatives and Differential Equations

  2. Differentiation • Differential change

  3. Derivative Definition

  4. Taylor Series

  5. Taylor Series Graphically

  6. Numerical Differentiation Based on Taylor Series • Forward finite-divided difference • Backward finite-divided difference • Centered finite-divided difference

  7. Forward Finite-Divided Difference Approximation

  8. Forward Finite-Difference

  9. Backward Finite-Divided Difference Approximation

  10. Backward Finite-Difference

  11. Centered Finite-Divided Difference Approximation

  12. Centered Finite-Difference

  13. Unequally Spaced Data • One way to calculated derivatives of unequally spaced data is to determine a polynomial fit and take its derivative at a point. • As an example, using a second-order Lagrange polynomial to fit three points and taking its derivative yields:

  14. Derivatives and Integrals for Data with Errors • Numerical differentiation amplifies data errors. • Solution: fit a smooth, differentiable function to the data and take the derivative of the function.

  15. Numerical Differentiation with MATLAB • MATLAB has two built-in functions to help take derivatives, diff and gradient: • diff(x) • Returns the difference between adjacent elements in x. Not the same size as vector x. • diff(y)./diff(x) • Returns the difference between adjacent values in y divided by the corresponding difference in adjacent values of x

  16. Numerical Differentiation, function of a single variable • fx = gradient(f, h) • Gradient can also be used to find partial derivatives for matrices: [fx, fy] = gradient(f, h)

  17. Numerical Differentiation, function of a two variables To generate the components of a derivative of a function of two variables x, y use [fx, fy] = gradient(f, h) Where h scales the magnitude of vectors displayed

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